Ideal Sliding Mode Cone in Control Theory

• Ideal sliding mode cone is an invariant conic region in state space that generalizes traditional sliding mode surfaces to ensure robust dynamics against uncertainties.
• It utilizes optimization methods such as SOS programming, convex cone analysis, and LMIs to design controllers with guaranteed finite-time convergence and enlarged invariant regions.
• Its applications span event-triggered control, nonlinear uncertain systems, and geometric minimal surface theory, providing a unified framework for robust and optimal control design.

An ideal sliding mode cone is a construct in control theory and geometric analysis that generalizes the classical idea of a sliding mode manifold, providing an invariant conic region or set in state space where the system trajectories possess robust, ideally invariant dynamics despite uncertainties or disturbances. The concept appears in both rigorous mathematical formulations for nonlinear uncertain systems and as a geometric object in minimal surface theory. In modern robust and optimal control, the ideal sliding mode cone encapsulates both the invariance principle central to sliding mode control (SMC) and the optimization of convergence and region of attraction, serving as a design and analysis tool for both continuous- and event-triggered schemes.

1. Theoretical Foundations and Definitions

The ideal sliding mode cone extends the traditional notion of a sliding surface (or manifold) to a generally higher-dimensional, possibly nonlinear or “conic” invariant set in the system state space. In the SMC context, the classical sliding manifold is typically a codimension-one surface (e.g., $S = \{x\,:\, c^\top x = 0\}$) on which the system dynamics become insensitive to matched perturbations. The ideal sliding mode cone generalizes this by considering not only a surface but a conic region—shaped by the geometry of multiple manifolds or the intersection of several constraints—within which the invariant “ideal” sliding motion is enforced, see (Jiang et al., 16 Sep 2025, Feng et al., 2018). Formally, the ideal cone can be constructed, for instance, by two rotated manifold boundaries:

$$\hat{S} = \{x: \hat{c}^\top x = 0\}, \quad \check{S} = \{x: \check{c}^\top x = 0\},$$

and their intersection defines the conic region $D = \{x : \hat{s}(t)\,\check{s}(t) \leq 0\}$ in which the reduced-order dynamics retain robust invariance.

In a geometric measure theory context, the notion of sliding minimal cones involves the minimization of surface energies under sliding boundary conditions, yielding a classification of minimal cones (tangent cones) whose geometry and contact angles with boundaries are precisely constrained by variational and calibration arguments (Cavallotto, 2018).

2. Invariant and Robust Dynamics in Control

In sliding mode control, the goal is to design a feedback law such that the system state is forced onto a pre-defined manifold; once on this manifold, the dynamics are governed by an “ideal” reduced system insensitive to matched disturbances (Imahe, 2017). The ideal sliding mode cone strengthens the invariance principle: the state is not only attracted to a surface but remains confined in a conic region where the convergence to equilibrium can be guaranteed asymptotically (or in finite time) in the presence of practical implementation limitations or under event-triggered sampling (Jiang et al., 16 Sep 2025).

In robust control of affine or nonlinear uncertain systems, this invariance is systematically achieved by using convex cone sets, sum-of-squares (SOS) optimization, or linear matrix inequalities (LMIs) to ensure both the existence of Lyapunov functions and the satisfaction of invariance and finite-time convergence properties (Sanjari et al., 2016, Coutinho et al., 15 Nov 2024, Feng et al., 2018). Theoretically, the design process involves the joint search over Lyapunov functions and sliding manifolds (often polynomials), with the cone (or conic region) defined by admissible sublevel sets or the solution to bilinear or convex constraints.

3. Methodologies and Optimization Approaches

The implementation of an ideal sliding mode cone in controller synthesis leverages optimization-based frameworks:

• Sum of Squares (SOS) Programming: Nonnegativity and stability requirements are posed as SOS constraints. An iterative algorithm jointly updates candidate sliding manifolds $S(z)$ and Lyapunov functions $V(z)$ by solving a sequence of SOS problems. Constraints like $V(z) - l_1(z_1) \in \Sigma_{n-m}$ ensure positive definiteness of $V$, while others enforce decay rates and region-of-attraction expansion (Sanjari et al., 2016).
• Convex Cone Analysis: In multi-input affine systems with uncertain, non-positive definite high-frequency gain matrices (HFGM), the solution to the sliding control law is formulated as finding a unique solution to a nonlinear vector equation over the union of convex cones, guaranteeing existence and uniqueness under relaxed uncertainty norms (Feng et al., 2018).
• LMIs and Convex Optimization: For polytopic uncertain systems, LMIs are solved for controller gains such that Lyapunov functions are strictly decreasing and finite-time global stability is achieved. Explicit optimization objectives (e.g., minimizing the reaching time or maximizing the region of attraction—the “ideal cone” of initial conditions) are included in the design (Coutinho et al., 15 Nov 2024).

These methods enable systematic, nonconservative controller synthesis, enlarging the invariant region (i.e., the “ideal cone”) and providing explicit performance guarantees.

4. Event-Triggered and Piecewise Continuous Control

In digital or hybrid systems, the ideal sliding mode cone is enforced via event-triggered mechanisms that monitor the size and direction of the error state. Rather than using a practical sliding mode band (a neighborhood around the sliding manifold), the event-triggered SMC design employs two rotated sliding manifolds defining the boundaries of a cone. The event-triggering logic uses angular deviation and state error thresholds to ensure that the state remains in the conic region, leading to asymptotic convergence to the origin rather than to a residual set (Jiang et al., 16 Sep 2025).

Mathematically, the cone’s angular width is characterized as

$$\theta = \pi - \arccos\left( \frac{\check{c}^\top \hat{c}}{\|\check{c}\|\,\|\hat{c}\|} \right),$$

with control events triggered when either the state vector rotates outside a permitted angular deviation or the state error exceeds a norm-weighted threshold, thus enforcing both direction and magnitude-based safety.

A trade-off arises: higher triggering frequency (and thus increased communication) is required as the state nears the equilibrium to achieve true asymptotic stability, but this can be relaxed for practical purposes by adopting a slightly larger “practical cone” (as opposed to the “ideal” one).

5. Geometric and Variational Characterizations

In geometric measure theory, sliding minimal cones are classified based on how their branches meet the sliding boundary under variational energy minimization with weighted contributions from portions on/off the sliding set (Cavallotto, 2018). For one-dimensional cones in half-planes and two-dimensional cones in half-spaces, the minimality is dictated by angle conditions such as $\cos\theta_{\alpha} = \alpha$, where $\alpha$ is the weight in the energy functional. Paired calibration and blow-up techniques are used for minimality proofs.

In geometric control, especially for systems evolving on non-Euclidean manifolds or Lie groups, the “ideal sliding mode cone” is interpreted as the invariant set (for example, a sliding subgroup in a Lie group tangent bundle) on which robust motion is achieved (Espindola et al., 2023, Castaños, 28 Apr 2025). Differential-geometric methods enable the definition of sliding surfaces/subgroups and the analysis of dynamics and invariance on nonlinear state spaces. Topological obstructions (e.g., on the Möbius bundle or 2-sphere) can influence, or even restrict, the existence and globality of such invariant sets.

6. Applications, Performance, and Illustrative Results

The ideal sliding mode cone has been validated in a range of practical and theoretical contexts:

• Nonlinear uncertain systems: Iterative SOS techniques yield sliding manifolds and Lyapunov functions that maximize the inner bound of the invariant region and demonstrate both asymptotic and finite-time stability (Sanjari et al., 2016).
• Multivariable and polytopic systems: Robust SMC laws derived via LMIs guarantee global finite-time stability; explicit regions of initial conditions (the “ideal cones”) are calculated and shown to cover large portions of state space (Coutinho et al., 15 Nov 2024).
• Event-triggered robust control: The ideal sliding mode cone concept enables asymptotic convergence in linear systems and quadrotor UAVs under event-triggered SMC, with rigorous guarantees of positive inter-event lower bounds and reduced communication requirements (Jiang et al., 16 Sep 2025).
• Sliding minimal cones: Classification of one- and two-dimensional minimal cones with sliding boundaries provides models relevant in geometric analysis and materials science (Cavallotto, 2018).
• Geometric and differential-geometric SMC: Extension of invariant set concepts to Lie groups and manifolds unites coordinate-free trajectory tracking with robust invariance against matched disturbances (Espindola et al., 2023, Castaños, 28 Apr 2025).

Representative summary table:

Domain	“Ideal Cone” Role	Methodology/Property
SMC (uncertain nonlinear/affine)	Invariant region	SOS, convex cone, LMIs
Event-triggered SMC	Conic confinement	Hybrid event rules
Minimal surfaces (geometry)	Sliding tangent cone	Energy/cos θ = α, calibration
Non-Euclidean/Geometric SMC	Invariant subgroup/cone	Differential geometry

7. Connections and Contemporary Research Directions

The concept of the ideal sliding mode cone offers a unifying analytic and design framework for robustness and invariance in both control theory and geometric analysis. Contemporary advances include:

• Joint SOS/LMI optimization to synthesize enlarged invariant cones with explicit convergence time tuning (Sanjari et al., 2016, Coutinho et al., 15 Nov 2024).
• Differential-geometric and quotient-based formalisms to generalize sliding mode invariance to manifold-valued state spaces, accounting for topological obstructions (Castaños, 28 Apr 2025).
• Application to advanced safety- and performance-critical scenarios such as sliding mode control barrier functions for systems with high relative-degree safety constraints, where robust invariance must be preserved despite high-order dynamics and uncertainties (Chinelato et al., 2020).

A plausible implication is that future work will expand these methodologies to distributed systems, systems with output constraints, and more general nonsmooth or hybrid dynamics, leveraging the ideal sliding mode cone as the foundational analytic object for robust, optimized, and invariant control design in complex dynamical systems.